(1) Suppose that {r, y, z} is a linearly independent subset of vectors in R". Show that {x - z + y, z – y + x, y – x + z} is also linearly independent. (ii) Let u, v and w be vectors in R such that {u+v, u+ w, v + w} is linearly independent. Does it necessarily follow that {u, v, w} is also linearly independent? (Hint: Put a = u + v, y = u+ w, z = v + w. Then by hypotheses, {r, y, z} is linearly independent. Observe that r - z + y = 2u and so forth and make use of part (i1).)

(1) Suppose that {r, y, z} is a linearly independent subset of vectors in R". Show that {x - z + y, z – y + x, y – x + z} is also linearly independent. (ii) Let u, v and w be vectors in R such that {u+v, u+ w, v + w} is linearly independent. Does it necessarily follow that {u, v, w} is also linearly independent? (Hint: Put a = u + v, y = u+ w, z = v + w. Then by hypotheses, {r, y, z} is linearly independent. Observe that r - z + y = 2u and so forth and make use of part (i1).)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.3: Spanning Sets And Linear Independence

Problem 21EQ

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